Retroreflector with controlled divergence made by the method of groove undulation

ABSTRACT

A ruling of cube corner elements comprising intersecting ruled vee-grooves is characterized in that at least one of the vee-grooves is made such the substrate being ruled and the cutting tool are oscillated with respect to one another during the ruling of the vee-groove. Rulings made in accordance with the instant invention can be designed to produce a cube corner element having broader divergence.

This patent application claims the benefit of U.S. ProvisionalApplication No. 60/297,394 filed Jun. 11, 2001.

BACKGROUND OF THE INVENTION

This invention relates to a method of making a retroreflective articlehaving controlled divergence, and articles made by the method.

It is well known that retroreflective articles can be made from an arrayof microcube corner elements. Such an array of microcube corner elementscan be made by ruling a master of “male” cube corners into a planarsurface of a plate. This is taught generally by Stamm U.S. Pat. No.3,712,706; and also is taught in detail in Pricone U.S. Pat. No.4,478,769. Each of these patents is incorporated herein by reference inits entirety.

U.S. Pat. No. 4,478,769 describes a well-known method of makingtriangular cube corner elements, in which the planar surface of a masterplate is ruled with a diamond cutting tool that cuts a series of preciseparallel vee-grooves. To rule equilateral triangular cube corners, threesets of parallel grooves intersecting one another at angles of 60° aremade; each groove also will have an included angle of substantially70.53° disposed symmetrically, and will be ruled to a groove depthdetermined by the height of the cube corners desired. This methodautomatically results in an array of pairs of oppositely orientedequilateral triangular microcubes on the face of the master. To rulenon-equilateral triangle cube corners the grooves within the parallelsets will contain angles other than 70.53°, and intersect at anglesother than 60°, as disclosed, for example in Rowland U.S. Pat. No.3,384,348. Methods for ruling non-triangle cube corners generally do notuse three sets of parallel symmetrically disposed vee-grooves, but thefaces of the cube corners are nevertheless formed from the walls ofgrooves, as disclosed, for example in Nelson U.S. Pat. No. 4,938,563.

The ruled master may then be used to make a series of duplicates, suchas by electroforming, and the duplicates are assembled together to forma single “mother” tool. The assembled “mother” tool is used toelectroform molds, which then can be assembled into a tool capable ofproviding microcube retroreflective elements on a web of plasticsheeting material such as by embossing, casting, or other methods knownin the art.

Microcube corner retroreflective sheeting such as made by the methoddescribed above is used in highway safety applications such as highwaysigns and pavement markers. In such applications, the microcube cornerelements retroreflect light from a vehicle's headlights back to thedriver of the vehicle. This is an inexact retroreflection in which thedivergence angle, α, ranges between approximately 0° and more than 3°.The value of α operative in any given situation depends on the geometryof the vehicle and the distance from the vehicle to the retroreflectivematerial. For example, the divergence angle α for a large truck's rightheadlight and its driver at a distance of about 40 meters from a roadsign will be approximately 3°, while the divergence angle α for anautomobile's left headlight and its driver at a distance of about 600meters from a road sign will be approximately 0.05°.

Also associated with the divergence angle, α, is a rotation angle, ε,which is a measure of the direction of the divergence. The value of εwill be different for left and right headlights of a vehicle, and willalso depend on the vehicle geometry and the position of the road sign.

Ideally, microcube corner retroreflective sheeting used in road signswill produce a pattern of retroreflected light having sufficientintensity over a range of divergence angle values and rotation anglevalues. For example, even a non-urban retroreflective highway signshould retroreflect light through a divergence angle α of about 1°,which corresponds to the value of α from a large truck's right headlightback to its driver at a distance of about 120 meters from the road sign.

Improvements in the precision with which microcube corner elements canbe ruled in a master plate and duplicated by embossing have led toconcerns that such microcube corner retroreflective sheeting may beretroreflective over only a very narrow range of divergence angle, suchas about 0.0–0.5 degrees, as well as narrow ranges of rotation angle. Itwould be preferred to provide a ruled array with cube corners producingthe entire desired range of divergence and within very short distanceson the ruled array.

Light that is retroreflected by micro-sized cube corner elements willexperience a certain amount of diffraction because of the very smallsize of the microcubes. Such diffraction will result in retroreflectionover broader ranges of both divergence angle and rotation angle. Theparticular ranges of α and ε will depend on the particular diffractionpattern of a given microcube, which will depend in turn upon the cubesize, cube shape, the index of refraction of the cube material, and uponwhether or not the cube faces have been metallized. Diffraction,however, is not a desirable method to enhance retroreflection throughbroader divergence and rotational angle, because the very smallmicrocubes that achieve greater diffraction also cause a substantialquantity of light to be retroreflected with a divergence angle α ofgreater than about 3°, where the light is not useful to the vehicledriver. This is summarized in Table 1.

Table 1 indicates the spreading of retroreflection due to diffraction.Acrylic equilateral triangle cube corners are used in each case. Themillimeter dimension measures the edge length of the triangle(identically 2.449×the cube depth, or 1.155×the ruling spacing). Thepercentages indicate how much of the total retroreflected flux is withina 1°, 2°, or 3° maximum observation angle. For example, for the trianglecube corner with side 0.05 mm only 27.9% of the total retroreflectedlight arrives between 0° and 1° observation angles.

TABLE 1 Diffraction spreading of retroreflection from different sizetriangle cube corners 0.4 mm 0.2 mm 0.1 mm 0.05 mm 0° to 1° 91.6% 82.5%66.7% 27.9% 0° to 2° 95.7% 91.6% 82.4% 66.6% 0° to 3° 97.1% 94.4% 88.9%79.1%

Diffraction results in idiosyncratic patterns which are unlikely todistribute the retroreflected light in a manner that will be most usefulto a vehicle's driver. This is shown in FIGS. 4A–D.

It is known in the art to create intentional aberrations in cube cornerelements by causing the dihedral angles of the cube corner elements todeviate slightly from 90°. The classic paper “Study of Light DeviationErrors in Triple Mirrors and Tetrahedral Prisms,” J. Optical Soc. Amer.,vol. 48, no. 7, pp. 496–499, July, 1958 by P. R. Yoder, Jr., describesthe well-known spot patterns resulting from such aberrations.

U.S. Pat. No. 3,833,285 to Heenan, assigned to the common assignee andincorporated herein by reference in its entirety, teaches that havingone dihedral angle of a macro-sized cube corner element greater than theother two results in extended observation angularity in macrocubes, andspecifically that the retroreflected light diverges in an elongatedpattern.

U.S. Pat. No. 4,775,219 to Appledorn discloses retroreflective articleshaving tailored divergence profiles, wherein the cube corner elementsare formed by three intersecting sets of parallel vee-grooves, andwherein at least one of the sets includes, in a repeating pattern, atleast two groove side angles that differ from one another.

U.S. Pat. No. 6,015,214 to Heenan et al., assigned to the commonassignee, teaches methods of forming microcubes by ruling vee-groovesinto the edges of a plurality of flat plates, and discloses that thetilt angle of a cutting tool with respect to the surface of the edgesbeing ruled can be adjusted continuously as each groove is cut as afunction of the distance traveled by the cutting tool along the ruledsurface.

It is thus one object of the invention to provide an article comprisingan array of retroreflective microcube corner elements having controlledbroader divergence.

It is another object of the invention to provide methods for making suchan article.

SUMMARY OF THE INVENTION

A retroreflective article having a controlled broader divergence isprovided by ruling one or more sets of generally parallel vee-grooves toform a plurality of cube corner elements, each vee-groove having twoside walls that intersect at a groove root, in which ruling non-uniformdeviations of the cube dihedral angles from exactly 90° areintentionally introduced by causing the cutting tool and the surface ofthe substrate to oscillate with respect to one another in a controlledmanner during the ruling of at least one of the vee-grooves. Thevee-groove so formed will be an undulating groove. The cube cornerelement dihedral angles having at least one face defined by a side wallof an undulating vee-groove will be non-orthogonal to varying extents,depending on the phase, frequency, and amplitude of the oscillationduring ruling. This introduction of variable, controllednon-orthogonality of dihedral angles within very short distances of eachother on a single ruled groove will result in a controlled broaderdivergence of the ultimate retroreflective article made from such ruledcube corner elements.

DESCRIPTION OF THE FIGURES

FIG. 1 illustrates a substrate oriented in the x-y plane and having avee-groove ruled in the y-direction using the method of the prior art;and

FIG. 2 illustrates a small portion of an array of cube corner elementsmade in accordance with the first described mode of practicing theinvention, in which the magnitude of the groove undulation and thedeviations of the dihedral angles are greatly exaggerated for the sakeof illustration.

FIG. 3 illustrates a small portion of an array of cube corner elementsmade in accordance with the third described mode of practicing theinvention, in which the magnitude of the groove undulation and thedeviations of the dihedral angles are greatly exaggerated for the sakeof illustration.

FIGS. 4A–D illustrate diffraction patterns from four different prior artunaberrated cube corners. The patterns show observation angle from 0° to3° over all rotation angles. The entrance angle is 0°. The illustrationsscale the pattern logarithmically so that one step corresponds toretroreflectance difference of approximately 2.5 times. FIG. 4A is foran uncanted acrylic triangle cube, 0.2 mm base dimension, 0.082 mmruling depth. FIG. 4B is for the identical cube corner when aluminized.FIG. 4C is for a −9.74° (face more parallel) canted acrylic trianglecube, 0.256 mm base, 0.091 mm ruling depth. FIG. 4D is for a +11.17°(edge more parallel) acrylic triangle cube, 0.194 mm base, 0.089 mmruling depth. Cube sizes have been chosen having equal optical activeareas for purpose of diffraction comparison.

FIG. 5 illustrates the calculated diffraction pattern for the prior artcube corner of FIG. 4A, but with a simple aberration of +14 arc minuteson each dihedral angle. The geometric spot pattern has an averagedivergence of 1.1°.

FIGS. 6A–B illustrate calculated diffraction patterns for the cubecorner of FIG. 4A, but with aberrations in accordance with the firstmode of the present invention. FIG. 6A uses a sinusoidal undulation oneach of three grooves sufficient to give the geometric pattern anaverage divergence of 1.1°. FIG. 6B uses a combination of a simpleaberration of 9 arc minutes on each dihedral angle with a sinusoidalundulation on each of three grooves sufficient to give the geometricpattern an average divergence of 1.1°.

FIGS. 7A–B illustrate calculated diffraction patterns for the cubecorner of FIG. 4A, but with aberrations in accordance with the firstmode of the present invention being applied unequally to the threegrooves. For FIG. 7A only one of the three grooves receives sinusoidalundulation. For FIG. 7B two of the three grooves receive sinusoidalundulation. For each of these designs, the geometric light pattern hasan average divergence of 1.1°.

FIG. 8 compares of two forms of undulation called “sin” and “sin±½sin²”.

FIG. 9 compares the calculated observation angularity of the prior artaberrationless triangle cube corner for FIG. 4A, with four triangle cubecorners aberrated in accordance with the instant invention, being thosefor FIG. 5, FIG. 6A, the “sin±½ sin²” variant of FIG. 6A, and FIG. 6B.

FIG. 10 illustrates the application of equations (1)–(3) to the firstdescribed mode of the invention.

FIG. 11 illustrates the application of equations (4)–(6) to the seconddescribed mode of the invention.

FIG. 12 illustrates the application of equations (10)–(12) to the thirdand fourth described modes of the invention.

DETAILED DESCRIPTION OF THE INVENTION

Cube corner retroreflective articles made by a process including thestep of ruling one or more sets of generally parallel vee-grooves in asurface with a vee-shaped cutting tool can be provided with a controlledbroader divergence by oscillating the cutting tool and the substratewith respect to one another in a controlled manner as the tool cuts oneor more vee-grooves. The controlled oscillation during ruling willresult in an undulating vee-groove producing a controlled variation inthe dihedral angles of the cube corner elements on either side of theundulating groove, which will controllably broaden the divergence of theultimate retroreflective article. Preferably, an entire desired range ofdivergence will be provided over very short increments of groove length.

The term “attitude” as used herein shall mean the orientation of adefined axis of a cutting tool relative to the substrate surface to beruled. The defined axis will begin at the tool tip and point generallyaway from the substrate surface.

The term “cube corner elements” as used herein includes those elementsconsisting of three mutually intersecting faces, the dihedral angles ofwhich are generally on the order of 90°, but not necessarily exactly90°.

The term “groove root” as used herein means the continuous curve definedby the motion of the point of the cutting tool under the surface beingruled. A “groove root” cut in accordance with the method of thisinvention can be either straight or wavy, depending on the mode ofoperation of the instant invention.

The term “groove angle” as used herein means the included angle,measured in a plane normal to the groove root, between the two walls ofthe groove cut into the surface by the cutting tool at any given pointalong the groove length.

The phrase “the cutting tool and the substrate oscillate with respect toone another,” and substantial equivalents as used herein, shall meanthat during the ruling of a vee-groove either the cutting tooloscillates with respect to the substrate, or the substrate oscillateswith respect to the cutting tool, or both the cutting tool and thesubstrate oscillate at the same time, so as to create an undulatingvee-groove.

The term “divergence” as used herein is the angle between a light raythat enters a retroreflecting element (e.g., a cube corner) and thelight ray after leaving that element.

FIG. 1 illustrates a perspective view of a substrate 20 having a planarsurface 22 on which vee-grooves can be ruled with a vee-shaped cuttingtool, such as is known in the art. As illustrated in FIG. 1, thesubstrate 20 is oriented with the surface 22 to be ruled disposed in thex-y reference plane of the x-y-z orthogonal reference system, whereinthe z direction is perpendicular to the surface 22. Throughout thispatent, the terms “the x-y plane,” “the x-z plane,” and “the y-z plane,”will mean the x-y reference plane, x-z reference plane, and y-zreference plane, as defined by the x, y, and z reference axes of FIG. 1.FIG. 1 further illustrates a typical vee-groove 24 of the prior artruled parallel to the y-axis, with the groove root 25 being a straightline at a constant z-depth relative to the surface 22, and the groovehaving constant included angle.

It is to be understood that for rulings on a non-flat substrate, themethods of this invention will require small modifications obvious toone skilled in the art. Such modifications are considered to be withinthe scope of this invention.

The instant invention will be described in terms of four modes ofoperation. It will be understood, however, that these four modes ofoperation are not necessarily mutually exclusive, and that two or moresuch modes of operation can be employed simultaneously. For ease ofunderstanding, the modes of operation are described and illustratedherein with respect to the simplest case in which the cube cornerelements being ruled are uncanted equilateral triangles. Theapplicability of the instant invention is not so limited, however, andthe methods of the invention can be applied to canted triangularrulings, and hexagonal and rectangular cube corner elements such as areillustrated and described in U.S. Pat. No. 5,914,813 (Smith et al.);U.S. Pat. No. 5,721,640 (Smith et al.); U.S. Pat. No. 4,938,563 (Nelsonet al.); and U.S. Pat. No. 4,895,428 (Nelson et al.). Further, while theundulations illustrated and described herein are most easily imagined assinusoidal, it will be appreciated that such sinusoidal undulations arenot a requirement of the instant invention. There is only therequirement that the undulations must be piecewise smooth and that thetool accelerations be such that they will not break the tool or ruin thecutting. A single groove can be made in several consecutive parts, eachof which includes some aspect of undulation.

By way of example and not by way of limitation, FIG. 8 illustrates twoforms of undulation curves suitable for use in the various modes ofoperation of the instant invention. The curve labeled “sin” is a perfectsinusoid. The curve labeled “sin±½ sin²” follows the function sin−½ sin²from 0 to π and then the function sin±½ sin² from π to 2π, and continuesto alternate between the two functions. The sin±½/sin² undulation, byhaving nearly flatted regions, gives more weight to the unaberratedcubes. In FIG. 8, the sin undulation was adjusted to 0.6*sin so as toproduce the same average geometric divergence as the sin±½/sin²undulation, in order to provide a more appropriate basis of comparison.It will be understood that the horizontal and vertical scales of FIG. 8require further adjustment to the dimensions of the cube corner ruling,such as shown in Table 2.

The two curves in FIG. 9 labeled “20[sin]” and “33[sin±½ sin²]” show thedifference in observation angularity for the sin and sin±½ sin²undulations illustrated in FIG. 8, respectively. Both undulations werechosen to give average geometric divergence of 1.1°. The bracketnotation used in this application for denoting undulatory designsdescribes the extent of undulation. “20[sin]” denotes sinusoidalundulation of such amplitude and pitch that the single such groovecontributes a maximum of 20 arc minutes of dihedral angle error in theruling. “33[sin±½ sin²]” denotes sin±½ sin² undulation of such amplitudeand pitch that the single such groove contributes a maximum of 33 arcminutes of dihedral angle error in the ruling. That both rulings producethe same 1.1° average geometric divergence in the acrylic product isexplained by the sin±½ sin² undulation having nearly flatted regions andthereby producing a greater proportion of nearly unaberrated cubes inthe population. Undulations useful for the present invention do not haveto be perfectly periodic and do not have to follow any explicitmathematical functions.

The four simplest modes of operation of the invention are explainedbelow. In the following discussion, the notation {dot over (δ)} shallmean the change in the rise or fall of each groove root introduced bythe first mode of operation of the instant invention; the notation{umlaut over (δ)} shall mean the change in the direction of the grooveroot within the x-y plane introduced by the second mode of operation ofthe instant invention; and the notation $\overset{...}{\delta}$shall mean the change in the half angles of the respective grooves asruled in accordance with both the third and fourth modes of operation ofthe instant invention.First Mode—Vertical Undulation

In a first mode of operation of the instant invention, the cutting toolis maintained at a constant attitude with respect to the substrate, andduring the ruling of at least one vee-groove the cutting tool and thesubstrate oscillate with respect to one another in a vertical direction,i.e., in a direction parallel to the z-axis of FIG. 1. This will resultin a vee-groove having a non-constant groove angle. The resulting grooveroot is a vertically undulating curve. The two-dimensional projection ofthe groove root in the x-y plane is a straight line. However, theintersection of the groove wall with the x-y plane is a horizontallyundulating curve. In accordance with the definition of the phrase “thecutting tool and the substrate oscillate with respect to one another” asset forth above, it will be appreciated that the same effect can beachieved either by holding the substrate in a fixed position, and movingthe tip of the cutting tool in a vertically undulating curve, whilemaintaining the attitude of the cutting tool constant, or by verticaloscillation of the substrate while the cutting tool is moving in astraight line, or by simultaneous out-of-phase movement in thez-direction of both the cutting tool and the substrate. The choice ofwhether to move the cutting tool, or the substrate, or both during thecutting of the vee-groove will depend upon the design and functionalityof the graver that controls the cutting tool and the fixture that holdsthe substrate.

A first, inconsequential, effect of this first mode of operation of theinstant invention is the introduction of intersection errors in apattern of ruled triangular cube corner elements. That is, even if theother two groove sets are formed entirely of straight grooves of theprior art, the vertically undulating groove root made in accordance withthe first mode of operation of the instant invention will not alwaysintersect the vertices of the other two groove sets at their exactpoints of intersection.

A second, consequential, effect of this first mode of operation of theinvention is the purposeful introduction of variations or “errors” inthe dihedral angles of cubes having a cube face formed by a side wall ofan undulating groove. The frequency of the oscillation will be such thatone period of oscillation spans several cube corner widths. Thus, for asingle triangular cube described in part by a segment of an undulatinggroove root, the groove root segment is essentially descending,essentially ascending, or essentially level. The cube dihedral angleterminating at or near the lower end of a groove root segment will beslightly more obtuse than had the groove root been level. Similarly, thecube dihedral angle terminating at or near the raised end of a grooveroot segment will be slightly more acute, than had the groove rootsegment been level. For substantially equilateral triangle cube corners,the change to the cube dihedral angle is the groove slope angle dividedby √6. Generally, where all three grooves that define a cube cornerelement are vertically undulating grooves, then for each such cubecorner element, the deviation in depth of each groove will affect thetwo dihedral angles defined in part by that groove side wall, with thetotal effect on the dihedral angles being very nearly additive.

FIG. 10 summarizes the aberrations due to the first mode when theundulation is long enough that curvature within a single cube isinsignificant. A male equilateral triangle cube corner is illustrated asformed by three grooves. Each of the grooves g₁, g₂, g₃ are shown risingby a corresponding angular amount {dot over (δ)}₁, {dot over (δ)}₂, {dotover (δ)}₃ in the direction indicated. If a {dot over (δ)} value isnegative, then the groove is instead falling. The three dihedral edgesare labeled with their angle errors, e₁; e₂; e₃, that is, theirdeviations from perfect 90°. The dihedral angle errors are given byapproximate equations (1)–(3). $\begin{matrix}{e_{1} \approx \frac{{\overset{.}{\delta}}_{3} - {\overset{.}{\delta}}_{2}}{\sqrt{6}}} & (1) \\{e_{2} \approx \frac{{\overset{.}{\delta}}_{1} - {\overset{.}{\delta}}_{3}}{\sqrt{6}}} & (2) \\{e_{3} \approx \frac{{\overset{.}{\delta}}_{2} - {\overset{.}{\delta}}_{1}}{\sqrt{6}}} & (3)\end{matrix}$

It will be appreciated that these equations require some adjustment fornon-equilateral triangle cube corners.

FIG. 2 roughly represents a portion of a single groove g ruled inaccordance with the first mode of operation of the invention, viewed asa projection in the x-y plane, wherein the depth of the cutting tool isvaried with respect to the plane of ruling. In this case, the tip of thecutting tool is rising as the tool moves along the groove g from left toright in the Figure. In each cube, dihedral angle d₁ terminates at ornear the lower end of the groove root segment, and dihedral angle d₂terminates at or near the raised end of a groove root segment. Then inall ten cubes illustrated the dihedral angles d₁ will be slightlygreater than 90°, and the dihedral angles d₂ will be slightly less than90°. It may be seen in FIG. 2 that the dihedral angles d₁ and d₂ neednot terminate exactly at groove root g, but can terminate near thegroove root. It also may be seen that the dihedral angle d₃ isunaffected by the undulation of groove root g.

Second Mode—Horizontal Undulation

In a second mode of operation, the cutting tool is maintained at aconstant attitude and a constant depth with respect to the substrate,and during the ruling of at least one vee-groove the cutting tool andthe substrate oscillate with respect to one another in a horizontaldirection sidewise to the direction of ruling. This will result in agroove of constant depth along the z-axis and substantially constantgroove angle, and wherein the groove root is an undulating curve in aplane parallel to the x-y plane. It will be appreciated that the sameeffect can be achieved by holding the substrate in a fixed position, andmoving the tip of the cutting tool in an undulating curve in the x-yplane, while maintaining the attitude of the cutting tool constant, orby horizontal oscillation of the substrate while the cutting tool movesin a straight line, or by simultaneous out-of-phase horizontal movementof both the cutting tool and the substrate. The choice of whether tomove the cutting tool or the substrate, or both, will depend upon thedesign and functionality of the graver that controls the cutting tooland the fixture that holds the substrate. Such choices will beunderstood by those skilled in the ruling arts. Further, where the aimof the cutting tool is constant, rather than tangent to the undulatingcurve, there will be a very slight variation in groove angle, but thisvariation will not have a significant effect on divergence for theamplitudes of groove undulations applicable to the instant invention.

A first, inconsequential, effect of this second mode of operation of theinstant invention is the introduction of intersection errors in apattern of ruled triangular cube corner elements. That is, even if theother two groove sets are formed entirely of straight grooves of theprior art, the horizontally undulating groove root made in accordancewith the second mode of operation of the instant invention will notalways intersect the vertices of the other two groove sets at theirexact points of intersection.

A second, consequential, effect of this second mode of operation of theinvention is that the three corner angles of a triangular cube cornerare modified. The dihedral angle of a triangle cube corner is made acuteif it meets a triangle corner at which the angle has been modified to besmaller. Similarly the dihedral angle of a triangle cube corner is madeobtuse if it meets a triangle corner at which the angle has beenmodified to be larger. For a substantially equilateral triangle cubecorner, the change to a dihedral angle is approximately equal to thechange to the corresponding triangle corner angle divided by √3. Thesechanges to the dihedral angles of the affected cube corners result in acube corner article having broader divergence.

FIG. 11 summarizes aberrations resulting from the second mode when theundulation is long enough that curvature within a single cube isinsignificant. A male equilateral triangle cube corner is illustrated asformed by three grooves. Each of the grooves forming the triangle isshown rotated by a corresponding angular amount {umlaut over (δ)}₁,{umlaut over (δ)}₂, {umlaut over (δ)}₃ in the direction indicated. If a{umlaut over (δ)} value is negative, then the groove is rotatedoppositely. The three dihedral edges are labeled with their angleerrors, e₁, e₂, e₃, that is, their deviations from perfect 90°: Thedihedral angle errors are then given by approximate equations (4)–(6).$\begin{matrix}{e_{1} \approx \frac{{\overset{¨}{\delta}}_{3} - {\overset{¨}{\delta}}_{2}}{\sqrt{3}}} & (4) \\{e_{2} \approx \frac{{\overset{¨}{\delta}}_{1} - {\overset{¨}{\delta}}_{3}}{\sqrt{3}}} & (5) \\{e_{3} \approx \frac{{\overset{¨}{\delta}}_{2} - {\overset{¨}{\delta}}_{1}}{\sqrt{3}}} & (6)\end{matrix}$It will be appreciated that these equations require some adjustment fornon-equilateral triangle cube corners.Third Mode—Rocking To and Fro

In a third mode of operation of the invention, at least one vee-grooveis ruled so that the movement of the tip of the cutting tool defines astraight groove root parallel to the x-y plane, and the attitude of thecutting tool oscillates within a plane containing the groove root. Thatis, the upper end of the cutting tool oscillates to and fro parallel tothe direction of the groove. The attitude of the tool with respect tothe substrate must be controlled as a function of the position of thecutting tool along the groove. In this third mode of operation, thecenter of oscillation is preferably the tip of the cutting tool. Thismode produces a groove having groove walls that undulate such that theincluded groove angle expands and contracts along the length of thegroove.

As the attitude of the cutting tool oscillates in accordance with thisthird mode of the invention, the included groove angle will varycontinuously according to the well-known adjustment technique for grooveangle. A vee cutter with included half-angle h, raked to angle R, cuts agroove of half-angle H, slightly larger than h, as given by equation(8). $\begin{matrix}{H = {\tan^{- 1}\left( \frac{\tan\mspace{11mu} h}{\cos\mspace{11mu} R} \right)}} & (8)\end{matrix}$Regardless of the direction of rake, consider R positive. Consider asmall variation, of magnitude much less than the rake itself, applied toR. The variation ΔR may be positive or negative. When the rake anglechanges to R+ΔR, the included half-angle H of the groove changes by anamount ΔH given by approximate equation (9).ΔH≈sin h×cos h×sin R×ΔR  (9)Thus a rake undulation in the vee cutter produces an approximatelyproportional undulation in the half-groove angle.

FIG. 3 illustrates ten cube corners along a portion of a groove thatnarrows from left to right. The groove walls produced in accordance withthis third mode of the invention will intersect the x-y plane inundulatory curves corresponding to the variations in included grooveangle. These variations of the groove angle along the length of thegroove will result in variations in the dihedral angles of the cubecorner elements defined in part by the groove side walls. For each ofthe cubes illustrated in FIG. 3, the two dihedral angles d₁ and d₂ areapproximately equal, while their averaged value decreases from cube tocube from left to right along the illustrated portion of groove g. Thedihedral angles d₃ remain unaffected. For substantially equilateraltriangle cube corners, increasing a groove's included angle has theeffect of increasing the cube dihedral angles formed in part from thatgroove by an amount equal to the groove's increase divided by 2√2.Similarly, reducing a groove's included angle reduces corresponding cubedihedral angles by the same 2√2 factor. For a substantially equilateraltriangle cube corner formed from two or more modified grooves, themodifications to the cube dihedral angles are very nearly sums of theeffects of the separate groove modifications. The purposely introducedvariations in cube dihedral angles will result in a cube cornerretroreflective article of broader divergence.

FIG. 12 summarizes the aberrations due to the third mode when theundulation is long enough that curvature within a single cube isinsignificant. A male equilateral triangle cube corner is illustrated asformed by three grooves. Each of the grooves forming the triangle aredesignated by their half-groove angle errors,${\overset{...}{\delta}}_{1},{\overset{...}{\delta}}_{2},{\overset{...}{\delta}}_{3},$that is, their deviations from perfect 35.26° The three dihedral edgesare labeled with their angle errors, e₁, e₂, e₃, that is, theirdeviations from perfect 90°. The dihedral angle errors are then given byapproximate equations (10)–(12). $\begin{matrix}{e_{1} \approx \frac{{\overset{...}{\delta}}_{3} + {\overset{...}{\delta}}_{2}}{\sqrt{2}}} & (10) \\{e_{2} \approx \frac{{\overset{...}{\delta}}_{1} + {\overset{...}{\delta}}_{3}}{\sqrt{2}}} & (11) \\{e_{3} \approx \frac{{\overset{...}{\delta}}_{2} + {\overset{...}{\delta}}_{1}}{\sqrt{2}}} & (12)\end{matrix}$It will be appreciated that these equations require some adjustment fornon-equilateral triangle cube corners.Fourth Mode—Rocking Side to Side

Whereas the foregoing third mode of operation can be understood as“forward-backward” oscillation of the attitude of the cutting tool, thefourth mode of operation of the invention uses “side-to-side”oscillation of the cutting tool. That is, the tip of the cutting toolstill defines a straight groove root parallel to the x-y plane, whilethe attitude of the cutting tool oscillates within planes perpendicularto the groove root. That is, the attitude oscillates transverse to thedirection of the ruling. In this method, while the magnitude of thegroove angle itself will not change along the length of the groove, theprogressive change in attitude of the cutting tool along the groove willresult in variations in the angle between the x-y plane and the groovewalls that produce the cube corner elements on either side of thegroove. The dihedral angles of the cube corner elements defined in partby the undulating groove surfaces will thereby be modified, similarly tothe third mode of operation of the invention. The identical equations(10)–(12) apply to this fourth mode. These errors in the dihedral angleswill in turn result in a cube corner retroreflective article of broaderdivergence.

Curvature

It is contemplated that the undulatory method of the instant inventioncan give deviations from flatness of the ruled cube corner faces on theorder of 0.01°. For the sizes of cube corners contemplated, this will bejust a small fraction of a wavelength of visible light. This effect willbe insignificant, particularly when the retroreflective article madefrom the cube corner array is sheeting, retroreflective fabric, ortraffic control devices.

A Difference Among the Modes

For either equations (1)–(3) (i.e., the first mode of operation) orequations (4)–(6) (i.e., the second mode of operation) it is seen thatequation (7) must hold.e ₁ +e ₂ +e ₃=0  (7)Thus, the first and second modes, applied as described, result in cubecorners with no net dihedral angle error. It is sometimes desireable tohave net dihedral angle error, as shown in Example 4 below. To producenet error by the first or second modes, it is a simple matter toincorporate a bias error beneath the errors produced by undulation. Forexample, the groove angles can be chosen different from the perfect70.53°, and these imperfect grooves can be subjected to vertical (firstmode) or horizontal (second mode) undulation. To determine the imperfectgroove angles desired for the bias, equations (10)–(12) may be solvedfor the groove errors in terms of the dihedral errors. This gives newequations (13)–(15). $\begin{matrix}{{\overset{...}{\delta}}_{1} \approx \frac{{- e_{1}} + e_{2} + e_{3}}{\sqrt{2}}} & (13) \\{{\overset{...}{\delta}}_{2} \approx \frac{{+ e_{1}} - e_{2} + e_{3}}{\sqrt{2}}} & (14) \\{{\overset{...}{\delta}}_{3} \approx \frac{{+ e_{1}} + e_{2} - e_{3}}{\sqrt{2}}} & (15)\end{matrix}$

Again, it will be appreciated that these equations require someadjustment for non-equilateral triangle cube corners. The${\overset{...}{\delta}}_{1},{\overset{...}{\delta}}_{2},{{\overset{...}{\delta}}_{3}.}$from equations (13)–(15) will serve as a biasing adjustment to thegroove angles which will then be undulated to produce the additionalerrors according to the first or second mode. When using the third orfourth modes, no separate biasing is necessary, since the biasesdiscovered from equations (13)–(15) can be parts of the${\overset{...}{\delta}}_{1},{\overset{...}{\delta}}_{2},{\overset{...}{\delta}}_{3}$of the undulation.

In each of the foregoing modes of operation of the invention, the resultwill be that the dihedral angles of the cube corner elements formed bythe grooves cut by the inventive method will differ from one cube cornerelement to the next. This non-equivalency of the cube corner elementswill broaden, in a controlled manner, the divergence of aretroreflective structure made from an array including thenon-equivalent cube corner elements.

It will be appreciated that the modes of operation described above canbe used in any combination. Where the cube corner elements are made fromthree intersecting sets of approximately parallel grooves, the groovevariations can be made in any number of grooves in a set, and in one,two, or three sets. If grooves in more than one set are varied, theresult will be, in general, a randomized combination of three dihedralangles in each cube.

Also each undulation can be a compound undulation, produced by acombination of any number of the basic four modes, or any other mode, ofoscillation. For example the graver can be oscillating both verticallyand horizontally at once (mode 1+mode 2), or it can be rockingto-and-fro and side-to-side at once (mode 3+mode 4), or any of the nineother logical combinations. In compound modes, the amplitude and lengthof undulations need not agree. In all such cases, the equation sets(1)–(3), (4)–(6), and (10)–(12) can be separately applied to obtain thedihedral angle errors for one cube, and the separate results added todetermine the dihedral angle errors of the compound mode undulation.

FIGS. 10–12 and corresponding equation sets (1)–(3), (4)–(6), and(10)–(12) show the effect of the different modes of the invention on theaberration of a single cube corner. The purpose of this invention is toenable the optical designer to produce an array with a great variety ofaberrated cube corners. Each triangle cube corner has six contiguousneighbors (two at each of its vertices) with its same orientation. Eachof these neighbors of the first cube corner will in general have quitedifferent aberrations from the first, because these neighbors will beformed from one groove in common with the first, which will have changedslightly, and two grooves not in common with the first, which can haveundulations out of phase with the undulations of the correspondinggrooves forming the first cube. Those skilled in the art will recognizethat not all geometrical light patterns are achievable by the presentinvention. For one thing, as shown in the Yoder paper, there are limitson the geometrical light distributions achievable by dihedralaberrations. For another, not all distributions of dihedral aberrationare achievable by the present invention. Any technique that randomlycombines three groove variations must include all “cross terms”, thatis, for any part of the groove 1 variation, any part of the groove 2variation, and any part of the groove 3 variation, these three partswill occur together in forming a cube corner somewhere in the array.This makes the invention more suitable for smooth spreadings of theretroreflection than for precise arrangements of the light, asillustrated in Examples 3–6 described below.

COMPARATIVE EXAMPLES

All of the following Examples 1–6 will be based on an uncanted acrylictriangle cube corner having 0.2 mm base dimension corresponding to 0.082mm ruling depth. Observation angles from 0° to 3° will be considered,while entrance angle is fixed at 0°. The light source is assumed to havethe spectral power distribution of CIE Illuminant A, corresponding to anincandescent lamp, and the detector is assumed to have the spectralsensitivity function CIE V(λ), corresponding to human photopic vision.

All of the example outcomes are results of optical calculation.Undulatory modelings are based on 1000 randomly generated cube corners.A shorthand has been employed in describing sinusoidal undulations:N[sin] will be understood as undulation of degree such that the singlesuch groove contributes up to N arc minutes to dihedral angle error.

Example 1 Prior Art

The first example is the perfect, unaberrated cube corner withretroreflected light pattern shown in FIG. 4A. The geometric lightpattern must have zero divergence, but the cube size introducesnoticeable diffraction, as summarized in Table 1. The observation angleaspect of the diffraction light pattern is also shown in FIG. 9 as thecurve labelled “aberrationless”. The curve in FIG. 9 is derived fromFIG. 4A.

The geometrical light pattern from a single cube corner consists of sixpunctal spots, as explained in the cited work of P. R. Yoder, Jr. Anunaberrated cube corner produces its six spots in coincidence. In thefollowing Examples 2–6 different techniques are used to aberrate cubecorners. For purposes of comparison, the aberrations in each of theseexamples are chosen so that the geometrical light pattern in each casehas average divergence approximately 1.1°, where the divergence ismeasured from the centroid of all the punctal spots. All illustrationsshow diffraction light patterns rather than geometrical patterns, asthis is what small cube corners really produce.

Example 2 Prior Art

Example 2 represents the simplest way known to the prior art to make anarray of aberrated cube corners having average geometric divergence of1.1°. Each dihedral angle of each cube corner is made 14 minutes greaterthan the perfect 90°. FIG. 5 shows the resulting diffraction pattern.Most of the energy is near the characteristic six punctal spots asdescribed above, which diffraction joins into a ring. FIG. 9 shows thering as a hump in the curve labeled 14,14,14, peaking at about 1.1°. Theaverage intensity at 1.1° observation angle is about eight times theintensity at 0° observation angle. A retroreflector of this kind wouldnot have road applications since it only functions well over such ashort range of observation angles that, at any distance, the full rangeof vehicles (from trucks to cars) could not benefit. It could havespecialized applications for instrumentation.

Examples 3–6 illustrate rulings that can be made in accordance with theinstant invention. The illustrative rulings have planned aberrationsthat result in broadened divergence profiles. In all these examples, theundulations are adjusted to give the same average geometric divergence,1.1°, as in prior art Example 2. FIGS. 6A and 6B illustrate smoothdivergence spreadings achieved with the rulings of Examples 3 and 4,while FIGS. 7A and 7B show some limited aiming capabilities achievedwith the rulings of Examples 5 and 6. Table 2 gives the ruling detailscorresponding to Examples 3–6.

TABLE 2 Ruling details using vertical undulation EXAMPLE SHORTHANDFIGURE FORM PITCH AMPLITUDE START 33[sin ± ½ sin²] sin ± ½ sin² 4 mm0.0097 mm random 3 20[sin] 6A sin 4 mm 0.0116 mm random 4 9 + 15[sin] 6Bsin 4 mm 0.0087 mm random 5 35[sin] on G3 7A sin 4 mm 0.0204 mm random 624[sin] on G1 & G2 7B sin 4 mm 0.0135 mm randomFIG. 9 illustrates the observation angularity achieved with the rulingsof Examples 1–4, as well as the 33[sin±½ sin²]

Example 3

For Example 3 sinusoidal vertical undulation is assumed according toMode 1 of this invention. Ruling dimensions are shown in Table 2 under20[sin]. FIG. 6A shows the resulting calculated diffraction pattern. Ithas a gentle central peak and is uncluttered with diffraction andaberration artifacts out to about 2° observation angle. This is thebeneficial result of the cube corners not being alike as they were inExamples 1 and 2. FIG. 9 shows the observation angularity of Example 3as curve 20[sin]. Its intensity at 1.1° is only 37% that of prior artExample 2 at 1.1°. However its intensity at 0° is 16 times that of priorart Example 2, and its intensity at 2° is 2.2 times that of prior artExample 2. Unlike Example 2, a retroreflector having the aberrated cubecorners of Example 3 is not limited to close applications, but performswell over a broad distance range.

Example 4

For Example 4 sinusoidal vertical undulation according to Mode 1 isassumed in combination with a bias of 9 arc minutes error on eachdihedral angle. That is, all the grooves are initially planned to make 9arc minutes of dihedral error and a sinusoidal undulation 15/20 as greatas that of Example 3 is superimposed on this. Ruling dimensions aregiven in Table 2 under 9+15[sin]. FIG. 6B shows the resultingdiffraction pattern, practically flat to about 1.5°. It must beappreciated that the diffraction pattern illustrations arelogarithmically scaled and one step corresponds to difference inretroreflectance of approximately 2.5 times. FIG. 9 shows that theobservation angularity of Example 4 is nearly flat to about 1.5°.Example 4 could function as a close distance retroreflector in roadapplications.

Example 5

Example 5 shows the effect of sinusoidal vertical undulation accordingto Mode 1 but assumed for only one set of grooves, G3, of the three setsof grooves G1, G2, G3 ruled to make triangular cube corners. Rulingdimensions are given in Table 2 under “35[sin] on G3.” Equations (1)–(3)show how this affects two of the three dihedral angles. FIG. 7A showsthe diffraction pattern. This example illustrates how the method canproduce patterns with directed observation angularity. This is thereason this Example 5 is not included among the curves of FIG. 9, whichare averages over all directions.

Example 6

Example 6 shows the effect of sinusoidal vertical undulation accordingto Mode 1 assumed for two of the three groove sets. Ruling dimensionsare given in Table 2 under “24[sin] on G1&G2.” Equations (1)–(3) showhow this affects two of the three dihedral angles. FIG. 7B shows thediffraction pattern. This example illustrates a different type ofdirected observation angularity that can be achieved with the method ofthe instant invention.

It will be appreciated that while Examples 3–6 use Mode 1, using thedimensions in Table 2, Mode 2 could have been used with differentdimensions to give identical results and Modes 3 or 4 to give almostidentical results.

Preferably, the entire desired range of divergence will be provided overa short increment of groove length, in order to avoid a spottyappearance. For retroreflective road sign applications there should beno patches larger than about 4 mm diameter visibly different fromadjacent patches. Practical undulatory methods are possible in which thegraver makes its entire cycle of change within 4 mm. For the first orsecond modes of the invention, typical groove undulations haveapproximately 10 micron amplitude over 4 mm cycle. Using the first orsecond modes of the instant invention, sinusoidal undulation on allthree of the grooves results in cube corner dihedral angle errors havinga distribution curve that can be described as resembling the shape of asmaller triangle mounted upon a larger trapezoid.

It may be seen that, in each single-mode of operation of the invention,the amplitude of the movement of either the cutting tool, or thesubstrate, or both, can be either constant or variable, both along thelength of a single groove and from one groove to the next, thusfacilitating controlled broadening of observation angularity anywhere onthe ruled substrate as desired by the ruling designer. When modes arecompounded, they need not agree in amplitude or length. Further, suchcontrolled variability can be introduced into one, two, or all three ofthe groove sets, in the ruling of triangular microcubes, and into someor all of the grooves in each groove set.

The rulings made by the methods described herein can be used to makeretroreflective products according to methods known in the art. Forexample, the ruled surface can be replicated through successivegenerations, and the replicates can be assembled together either with orwithout replicates of unaberrated cube corner arrays. Seamless copies ofthe assembly can be made such as by electrodeposition of nickel toprovide tools of uniform thickness. Tools having such patterns of cubecorner elements can be used to manufacture retroreflective products suchas sheeting. Such manufacturing methods are known in the art andinclude, for example, embossing, casting, and compression molding. Thetool of the instant invention can be used in each of these manufacturingmethods and variations thereof.

The invention has been described herein in terms of preferredembodiments and methodologies. Those skilled in the art will recognizethat variations can be made to the embodiment disclosed herein withinthe scope of the invention.

1. A method of making a pattern of cube corner elements on a substratesurface comprising the step of ruling a plurality of vee-grooves intothe substrate surface using a cutting tool, said substrate surface lyingin the x-y plane of an orthogonal x-y-z reference system, wherein thecutting tool and the substrate surface are oscillated with respect toone another during the ruling of at least one of said vee-grooves. 2.The method of claim 1 wherein said cutting tool and said substratesurface are oscillated with respect to one another in the z directionduring said ruling of said at least one of said vee-grooves.
 3. Themethod of claim 1 wherein said cutting tool and said substrate areoscillated with respect to one another in the x-y plane sidewise to thedirection of the groove during said ruling of said at least one of saidvee-grooves.
 4. The method of claim 1 wherein at least one vee-groove isruled so that the movement of the tip of the cutting tool defines astraight groove root parallel to the x-y plane, and the attitude of thecutting tool oscillates within a plane containing the groove root. 5.The method of claim 1 wherein at least one vee-groove is ruled so thatthe movement of the tip of the cutting tool defines a straight grooveroot parallel to the x-y plane, and the attitude of the cutting tooloscillates within planes perpendicular to the groove root.
 6. The methodof claim 1 wherein at least one vee-groove is ruled so that the movementof the cutting tool relative to the substrate produces a compoundundulation.
 7. An article made by the method of claim 1, the articlecomprising a substrate surface lying parallel to the x-y plane of theorthogonal x-y-z reference system and having a pattern of cube cornerelements defined by intersecting vee-grooves formed by the ruling of thecutting tool, wherein each vee-groove comprises two groove side wallsintersecting at a groove root; and wherein at least one of thevee-grooves is an undulating groove caused by the cutting tool and thesubstrate surface being oscillated with respect to one another duringthe ruling of this vee-groove.
 8. The article of claim 7 wherein saidgroove root of said undulating groove is a curve undulating in a planeperpendicular to the x-y plane.
 9. The article of claim 7 wherein saidgroove root of said undulating groove is a curve undulating in a planeparallel to the x-y plane.
 10. The article of claim 7 wherein saidgroove root of said undulating groove is a straight line parallel to thex-y plane and said groove walls of said undulating groove undulate suchthat the included groove angle expands and contracts along the length ofthe groove.
 11. The article of claim 7 wherein said groove root of saidundulating groove is a straight line parallel to the x-y plane and saidgroove walls of said undulating groove undulate such that the includedgroove angle is substantially constant along the length of the groove.12. The article of claim 7 wherein said undulating groove has a compoundundulation within the x-y-z reference system.
 13. A method of making aretroreflective article comprising a. making a pattern of cube cornerelements on a substrate surface by ruling a plurality of vee-groovesinto the substrate surface using a cutting tool, said substrate surfacelying in the x-y plane of an orthogonal x-y-z reference system, whereinthe cutting tool and the substrate surface are oscillated with respectto one another during the ruling of at least one of said vee-grooves; b.making a replica tool from said ruling, said replica tool bearing saidpattern of cube corner elements; and c. using said tool to form aretroreflective article comprising a replica of said pattern of cubecorner elements.
 14. The method of claim 13 wherein said cutting tooland said substrate surface are oscillated with respect to one another inthe z direction during said ruling of said at least one of saidvee-grooves.
 15. The method of claim 13 wherein said cutting tool andsaid substrate are oscillated with respect to one another in the x-yplane sidewise to the direction of the groove during said ruling of saidat least one of said vee-grooves.
 16. The method of claim 13 wherein atleast one vee-groove is ruled so that the movement of the tip of thecutting tool defines a straight groove root parallel to the x-y plane,and the attitude of the cutting tool oscillates within a planecontaining the groove root.
 17. The method of claim 13 wherein at leastone vee-groove is ruled so that the movement of the tip of the cuttingtool defines a straight groove root parallel to the x-y plane, and theattitude of the cutting tool oscillates within planes perpendicular tothe groove root.
 18. The method of claim 13 wherein at least onevee-groove is ruled so that the movement of the cutting tool relative tothe substrate produces a compound undulation.
 19. A retroreflectivearticle made by the method of claim 13, said retroreflective articlecomprising the replica of said pattern of cube corner elements, wherebysaid replica comprises intersecting vee-grooves, wherein each vee-groovecomprises two groove side walls intersecting at a groove root, andwherein at least one of the vee-grooves is an undulating groove.
 20. Theretroreflective article of claim 19, wherein said groove root of saidundulating groove is a curve undulating in a plane perpendicular to thex-y plane.
 21. The retroreflective article of claim 19, wherein saidgroove root of said undulating groove is a curve undulating in a planeparallel to the x-y plane.
 22. The retroreflective article of claim 19,wherein said groove root of said undulating groove is a straight lineparallel to the x-y plane and said groove walls of said undulatinggroove undulate such that the included groove angle expands andcontracts along the length of the groove.
 23. The retroreflectivearticle of claim 19, wherein said groove root of said undulating grooveis a straight line parallel to the x-y plane and said groove walls ofsaid undulating groove undulate such that the included groove angle issubstantially constant along the length of the groove.
 24. Theretroreflective article of claim 19, wherein said undulating groove hasa compound undulation within the x-y-z reference system.